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We construct faces of the convex set of all $2\otimes 4$ bipartite separablestates, which are affinely isomorphic to the simplex $\Delta_{9}$ with tenextreme points. Every interior point of these faces is a separable state whichhas a unique decomposition into 10 product states, even though ranks of thestate and its partial transpose are 5 and 7, respectively. We also note thatthe number 10 is greater than $2\times 4$, to disprove a conjecture on thelengths of qubit-qudit separable states. This face is inscribed in thecorresponding face of the convex set of all PPT states so that sub-simplices$\Delta_k$ of $\Delta_{9}$ share the boundary if and only if $k\le 5$. Thisenables us to find a large class of $2\otimes 4$ PPT entangled edge states withrank five.

Geometry for separable states and construction of entangled states with positive partial transposes